On the noncommutative Poisson geometry of certain wild character varieties

Abstract

To show that certain wild character varieties are multiplicative analogues of quiver varieties, Boalch introduced colored multiplicative quiver varieties. They form a class of (nondegenerate) Poisson varieties attached to colored quivers whose representation theory is controlled by fission algebras: noncommutative algebras generalizing the multiplicative preprojective algebras of Crawley-Boevey and Shaw. Previously, Van den Bergh exploited the Kontsevich–Rosenberg principle to prove that the natural Poisson structure of any multiplicative quiver variety with tautological coloring is induced by an $H_0$-Poisson structure on the underlying multiplicative preprojective algebra; indeed, it turns out that this noncommutative structure comes from a Hamiltonian double quasi-Poisson algebra constructed from the quiver itself. In this article, we conjecture that, via the Kontsevich–Rosenberg principle, the natural Poisson structure on each colored multiplicative quiver variety is induced by an $H_0$-Poisson structure on the underlying fission algebra which, in turn, is obtained from a Hamiltonian double quasi-Poisson algebra attached to the colored quiver. We study some consequences of this conjecture and we prove it in two significant cases: the interval and the triangle.